7  Transport

Having established how single electrons in a band respond to external forces, we now move to the collective transport behavior of solids. The key new ingredients are statistics and distributions.

7.1 Recap – the Drude model

This is the simplest model for charge conduction (Drude, 1900), well known to all students after any electrodynamics course. It treats the electrons as a classical gas of free particles: each electron accelerates under the applied field and undergoes collisions (originally, with the metal cations) at an average rate \(1/\tau\), which relax its average momentum to zero in the form of a linear friction term. All these assumptions are quite drastic, yet the model works pretty well in describing many phenomena in conductors. We recall here key results, successes and clarify why/how we are unhappy with it.

import { renderDrudeCartoon } from "./_interactive/ch07-drude-cartoon.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;position:relative;margin:0 auto;";
  yield container;
  renderDrudeCartoon(container);
}

Figure 7.1: The Drude picture: a free electron bouncing off the metal cations along straight free flights, with specular collisions.

The equation of motion for the average momentum \(\langle\mathbf{p}\rangle\) of an electron in an electric field \(\mathbf{E}\) is

\[\frac{d}{dt}\langle\mathbf{p}\rangle = -\frac{1}{\tau}\langle\mathbf{p}\rangle + \mathbf{F}, \tag{7.1}\]

where the first term is a linear friction model linked to a scattering time \(\tau\), and often we use a force \(\mathbf{F}=q\mathbf{E}\) with \(q=-e\). This could be easily extended to magnetic fields by adding the Lorentz force. We can directly solve this linear model at finite frequency: for an oscillating field \(\mathbf{E}(t) = \mathbf{E}_0\,e^{-i\omega t}\), the ansatz \(\mathbf{p}(t) = \mathbf{p}_0\,e^{-i\omega t}\) turns Equation 7.1 into an algebraic relation, \((-i\omega + 1/\tau)\,\mathbf{p}_0 = q\mathbf{E}_0\). Given \(\mathbf{J} = nq\mathbf{v}\) with \(\mathbf{v}=\mathbf{p}/m\) this yields the law for AC Drude conductivity

\[\mathbf{J} = \sigma(\omega)\mathbf{E} = \frac{\sigma_0}{1-i\omega\tau}\mathbf{E},\qquad \sigma_0 = \frac{ne^2\tau}{m}\]

where \(\sigma_0\) is the familiar DC Drude conductivity and we can also introduce the electron mobility \(\mu_e=e\tau/m\) connecting the limit velocity with the electric field

\[\mathbf{v} = \frac{\mathbf{p}}{m} = -\frac{e\tau}{m}\mathbf{E} = -\mu_e\mathbf{E}, \tag{7.2}\]

which is also linked to conductivity by \(\sigma_0=ne\mu_e\). At finite frequency, the response is controlled by the product \(\omega\tau\):

• for \(\omega\tau \ll 1\) collisions keep the electrons in phase with the field and the response is dissipative, friction dominates and \(\sigma\) behaves as a real constant;
• for \(\omega\tau \gg 1\) the electrons oscillate almost freely according to their inertia, the current lags the field by \(90°\) and the response becomes mostly reactive (imaginary \(\sigma\)), with \(\sigma(\omega)\) decaying as \(1/\omega\).

Given that \(\tau\) for metals at room temperature falls in the 10s of femtoseconds, the crossover occurs around the far infrared region of the spectrum, i.e. well above anything related to electronics.

CautionDrude successes: \(\sigma(\omega)\) and plasma frequency

The link between conductivity and optical response should be a known fact from electrodynamics and comes from the fourth Maxwell equation \(\nabla\times\mathbf{H}=\mathbf{J}+\partial\mathbf{D}/\partial t\): for a monochromatic field \(\propto e^{-i\omega t}\) the right-hand side reads

\[\sigma\mathbf{E}-i\omega\varepsilon_0\mathbf{E} = -i\omega\varepsilon_0\left[1+\frac{i\sigma(\omega)}{\omega\varepsilon_0}\right]\mathbf{E},\] so the effect of free carriers can be absorbed into an effective relative dielectric function

\[\varepsilon_r(\omega) = 1 + \frac{i\sigma(\omega)}{\varepsilon_0\omega} = 1 - \frac{\omega_p^2}{\omega^2 + i\omega/\tau}\to 1-\frac{\omega_p^2}{\omega^2}, \tag{7.3}\]

where the limit on the right is taken for \(\omega\tau\gg 1\). In the equation, we defined the plasma frequency

\[\omega_p = \sqrt{\frac{ne^2}{m\varepsilon_0}} = \sqrt{\frac{\sigma_0}{\tau\varepsilon_0}}, \tag{7.4}\]

which typically falls in the UV spectral region for common metals (not for all conductors, e.g. consider the ionosphere where it can fall in the MHz range). In the weak-damping limit (\(\omega\tau\gg1\)) the dielectric function crosses zero at \(\omega = \omega_p\): below \(\omega_p\), the refractive index \(n+i\kappa=\sqrt{\varepsilon_r}\) becomes purely imaginary and reflectance

\[R = \left|\frac{1-(n+i\kappa)}{1+(n+i\kappa)}\right|^2,\]

is equal to 1 (approximate result: one typically always has some imaginary \(\varepsilon\)): this is why metals are shiny.

import { renderDrudeResponse } from "./_interactive/ch07-drude.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderDrudeResponse(container);
}

Figure 7.2: Drude model optical response. Left: AC conductivity \(\sigma(\omega)\) (real and imaginary parts). Center/Right: dielectric function \(\varepsilon(\omega)\) and reflectance \(R\), with shaded spectral region of transparency. The threshold between the two is defined by \(\mathrm{Re}[\varepsilon]=0\), or \(n=\kappa\). Note that we always have \(\omega_p\tau\gg 1\) in common metals.

Missing effects. Beware this is a quite simplified picture: the free-carrier (intraband) response of Equation 7.3 is not the whole story; interband transitions add their own contribution to \(\varepsilon(\omega)\), producing extra absorption and shifting the effective reflection edge. This is, for instance, what gives copper and gold their color; more on this when discussing optical properties.

CautionDrude failures: specific heat, Seebeck

Why are we unhappy with Drude? In practice, the functional form of the model (many parameters are just phenomenological, see comment on scattering below) generally provides a reasonably accurate description of charge transport, even if the exact connection with band structures is lacking. We will see that this can lead to subtle variations, for instance \(\sigma\) in general will be a tensor. These details are however typically not obvious to highlight experimentally.

Drude model fails badly for everything concerning thermal properties and thermal transport, starting from the electronic specific heat, that for a classic gas is expected to be

\[c_v = \frac{3}{2}nk_B,\]

due to classic equipartition for the three mechanical degrees of freedom of electrons. In practice, this number is off by a couple of orders of magnitude at room temperature and, unlike the constant classical prediction, the measured electronic contribution is strongly temperature dependent (linear in \(T\)). In connection to this, also thermal conductivity is wrongly described by Drude model. All these thermal failures originate from a single cause: Drude model came before quantum mechanics and completely neglects Pauli exclusion principle and thus the fermionic nature of electrons.

Origin of scattering. This is not a true failure of the model but a problem with the meaning of its parameters. In the original Drude model, scattering was assumed to come from collisions with metal cations. This was a reasonable assumption before knowing about quantum mechanics and Bloch states. In practice, electrons scatter on deviations from exact periodicity, including in particular defects and phonons, i.e. mechanical oscillations of the atomic network. Since \(\tau\) is basically a phenomenological parameter, this does not really affect the validity of the model.

7.2 The Sommerfeld model

The Sommerfeld model (1928) retains the free-electron picture of Drude, i.e. parabolic dispersion \(E(\mathbf{k}) = \hbar^2k^2/(2m^*)\) and collisions parametrized by \(\tau\), but replaces Maxwell–Boltzmann statistics with Fermi–Dirac statistics:

\[f(E) = \frac{1}{e^{(E-\mu)/(k_BT)}+1}. \tag{7.5}\]

This single change resolves the quantitative failures of the Drude model.

import { renderSommerfeldWell } from "./_interactive/ch07-sommerfeld-well.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;position:relative;margin:0 auto;";
  yield container;
  renderSommerfeldWell(container);
}

Figure 7.3: From the real crystal to the Sommerfeld model. Left: periodic atomic potentials with the allowed energy bands (gray), the Fermi level and the surface barrier rising to the vacuum level. Right: the effective square-well picture — a flat-bottom potential at \(E_0\) filled up to \(E_F\), with work function \(W\).

It is useful to consider what are the scales of key parameters in the Sommerfeld model. One normally distinguishes core electrons, in fully occupied bands, from valence electrons, assumed to be free to propagate in the metal. The mass will be a parameter \(m^*\), normally close to the free-electron one \(m_e\).

DefinitionOrders of magnitude in the Sommerfeld model

Here we consider aluminum (Al), not exactly an average metal, since it has many free electrons per unit cell and a compact crystal structure. Nevertheless, most metals fall not too far off. Al has three valence electrons and is an FCC crystal with \(a\approx 4\) Å, thus we have an electron density \(n=3\times4/a^3\approx 1.8\times10^{23}\,{\rm cm^{-3}}\). Assuming \(T=0\) we can easily calculate a Fermi wavevector \(k_F\)

\[n = \frac{2}{(2\pi)^3}\frac{4\pi}{3}k_F^3 = \frac{k_F^3}{3\pi^2} \implies k_F = \sqrt[3]{3\pi^2n}\approx 1.8\times 10^{10}\,{\rm m^{-1}}\]

and Fermi energy \(E_F\) and Fermi temperature \(T_F\), defined by \(E_F=k_BT_F\) and quantifying how close we are to the \(T\to0\) limit of the Fermi-Dirac distribution,

\[E_F = \frac{\hbar^2k_F^2}{2m_e}\approx 1.9\times 10^{-18}\,{\rm J} \approx 11.7\,{\rm eV},\qquad T_F = \frac{E_F}{k_B} \approx 1.35\times 10^5\,{\rm K}. \tag{7.6}\]

Given the melting temperature of Al is about \(1000\,{\rm K}\) (again not an average value, but the pure metal top value is reached by tungsten, with about \(3700\,{\rm K}\)), the Fermi-Dirac distribution for electrons in Al will always be very close to the unit step obtained in the \(T\to0\) limit.

CautionThe \(r_s\) adimensional parameter, aka Wigner-Seitz radius

Density \(n\) can be recast into an average volume per electron, expressed as a multiple of a sphere with the Bohr radius \(a_B\approx 0.5\) Å. This leads to the introduction of the parameter \(r_s\)

\[ \frac{1}{n} = \frac{4\pi}{3}(r_sa_B)^3 \]

where \(r_s\) falls in the \(2\)–\(6\) range for most metals. Using \(k_F = \sqrt[3]{3\pi^2 n}\), the Fermi wavevector is

\[k_F = \frac{(9\pi/4)^{1/3}}{r_s\,a_B} \approx \frac{1.92}{r_s\,a_B},\]

i.e. \(1/k_F\) is always a couple of Bohr radii and the Fermi wavelength \(\lambda_F = 2\pi/k_F \approx 3.3\,r_s\,a_B\) is a few Å, comparable to the lattice spacing. The Fermi energy is then most naturally measured in Rydbergs, \(\mathrm{Ry} = \hbar^2/(2m_ea_B^2) = \tfrac{1}{2}\alpha^2 m_ec^2 = 13.6\,{\rm eV}\):

\[E_F = \frac{\hbar^2k_F^2}{2m_e} = \frac{1.92^2}{r_s^2}\,{\rm Ry} \approx \frac{50}{r_s^2}\;\mathrm{eV}, \qquad T_F = \frac{E_F}{k_B} \approx \frac{5.8\times10^{5}}{r_s^2}\;\mathrm{K},\]

and the Fermi velocity comes out in units of the orbital velocity of the first Bohr orbit, \(\hbar/(m_ea_B) = \alpha c\)

\[v_F = \frac{\hbar k_F}{m_e} = \frac{1.92}{r_s}\,\alpha c,\]

where \(\alpha \approx 1/137\) is the fine-structure constant.

Over the metallic range \(r_s = 2\)–\(6\) these give \(E_F \approx 12.5\)–\(1.4\) eV, \(T_F \approx 1.5\times10^{5}\)–\(1.6\times10^{4}\) K and \(v_F \approx (2.1\)–\(0.7)\times10^{6}\) m/s: electrons cruising at a sizeable fraction of a percent of the speed of light at zero temperature, purely by Pauli exclusion. The aluminum numbers of the previous box correspond to \(r_s \approx 2.07\).

7.2.1 Metal = crystal with a Fermi surface

Any electron transport integral we perform on a metal is largely dominated by a thin \(k_BT\)-wide energy shell around the Fermi surface. The following key mathematical tool allows tackling this regime.

Theorem 7.1: The Sommerfeld expansion

Let \(\Gamma(E)\) be a smooth function describing whatever energy-dependent property of a metal. Let us assume that \(\Gamma(E)\) vanishes below some threshold and grows at most as a power law of \(E\). Then the thermal average

\[I(T) = \int_{-\infty}^{+\infty} \Gamma(E)\,f(E)\,dE,\]

with \(f\) the Fermi–Dirac distribution at chemical potential \(\mu(T)\), admits the low-temperature expansion

\[I(T) = \int_{-\infty}^{\mu}\Gamma(E)\,dE + \frac{\pi^2}{6}(k_BT)^2\,\Gamma'(\mu) + \mathcal{O}(T^4). \tag{7.7}\]

The leading explicit finite-temperature correction is quadratic in \(T\) and controlled by the slope \(\Gamma'(\mu)\) at the chemical potential. The expansion could be extended to \(T^4\) and further, but this is rarely cited. Note that there is an implicit \(T\) dependence in \(\mu(T)\), appearing in both terms. Since the expansion only makes sense in the limit of a metal, \(\mu(T)\) will always be close to its zero temperature limit, \(E_F\), with minor \(T\)-dependent variations.

Show proof

Integrating by parts with the primitive \(G(E) = \int_{-\infty}^{E}\Gamma(E')\,dE'\) gives

\[I(T) = \left.G(E)\,f(E)\,\right|_{-\infty}^{+\infty} + \int_{-\infty}^{+\infty} G(E)\left(-\frac{\partial f}{\partial E}\right)dE.\]

The hypotheses above imply the boundary term vanishes: at \(E \to -\infty\), \(G \to 0\) because \(\Gamma\) vanishes below some threshold; at \(E \to +\infty\), \(f\) decays exponentially while \(G\) grows at most like a power of \(E\). Hence

\[I(T) = \int_{-\infty}^{+\infty} G(E)\left(-\frac{\partial f}{\partial E}\right)dE.\]

Here \(-\partial f/\partial E\) is an even, normalized peak of width \(\sim k_BT\) centered at \(\mu\): we Taylor-expand \(G(E)\) around \(\mu\)

\[ G(E) = G(\mu) + \left.\frac{dG}{dE}\right|_\mu (E-\mu) + \frac{1}{2!} \left.\frac{d^2G}{dE^2}\right|_\mu (E-\mu)^2+\cdots \]

import { renderSommerfeldExpansion } from "./_interactive/ch07-sommerfeld-expansion.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;position:relative;margin:0 auto;";
  yield container;
  renderSommerfeldExpansion(container);
}

Figure 7.4: Graphical Taylor expansion of the primitive \(G(E)\) against the thermal weight \(-\partial f/\partial E\) (width \(\propto k_BT\)): by parity, only the even orders contribute.

Odd powers of \((E-\mu)\) integrate to zero by symmetry and, by variable change it is easy to show that any term \((E-\mu)^{2n}\) will scale as \((k_BT)^{2n}\). The quadratic term involves the integral

\[\int_{-\infty}^{+\infty} x^2\,\frac{e^x}{(e^x+1)^2}\,dx = \frac{\pi^2}{3},\]

leaving exactly Equation 7.7. \(\blacksquare\)

CautionApplication – Electronic specific heat

Chemical potential. We first apply Theorem 7.1 to the integral \(N=\int D(E)f(E)dE\), to track \(\mu(N,T)\) at fixed \(N\). We have

\[ N = \int_{-\infty}^\mu D(E)dE + \frac{\pi^2}{6}(k_BT)^2D'(\mu) + \mathcal{O}(T^4) \]

whose derivative in \(T\) yields

\[\frac{dN}{dT} = 0 = D(\mu)\frac{d\mu}{dT} + \frac{\pi^2}{3}k_B^2T\,D'(\mu) + \frac{\pi^2}{6}(k_BT)^2D''(\mu)\frac{d\mu}{dT} + \mathcal{O}(T^3)\]

and finally, up to leading order,

\[\frac{d\mu}{dT} = - \frac{\pi^2}{3}k_B\frac{D'(\mu)k_BT}{D(\mu)} + \mathcal{O}(T^3).\]

Typical interpretation: \(\mu(T)\) tends to move away from regions of high density of states \(D(E)\). This is perfectly reasonable: with growing \(T\), the Fermi-Dirac tails are extending in energy and probing states above and below \(E_F\); depending on which side hosts more states, \(\mu(T)\) has to adjust to keep a constant integral number of states.

Specific heat. Estimating now \(U = \int ED(E)f(E)dE\) we get

\[ \begin{aligned} \frac{dU}{dT} = c_v &= \frac{d}{dT}\left\{\int^\mu ED(E)dE + \frac{\pi^2}{6}(k_BT)^2\left[D(\mu)+\mu D'(\mu)\right]\right\} + \mathcal{O}(T^3)\\ &= \frac{\pi^2}{3}k_B\cdot D(\mu)k_BT + \mathcal{O}(T^3), \end{aligned} \]

which has an interesting meaning: in the classical limit we have \(c_v=3nk_B/2\); now, neglecting the exact numerical factor in front, \(n\) is replaced by \(D(\mu)k_BT\), i.e. the number of electrons inside the \(k_BT\) shell around the Fermi surface. In fact, these are the only electrons that are really able to take up thermal energy. So the result is fundamentally similar, only the number of truly active particles has changed due to Pauli exclusion.

import { renderMBvsFD } from "./_interactive/ch07-mb-vs-fd.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;position:relative;margin:0 auto;";
  yield container;
  renderMBvsFD(container);
}

Figure 7.5: Comparison between the heating of a classical electron gas and one dominated by Pauli exclusion. Note the total kinetic energy of electrons on the right hand side is much larger, but the number of electrons able to take thermal energy is limited by the statistics.

The prefactor is called the Sommerfeld coefficient

\[\gamma \equiv \frac{\pi^2}{3}\,k_B^2\,D(E_F), \tag{7.8}\]

directly measurable from the low-temperature linear term in the heat capacity: it provides experimental access to the density of states at the Fermi level.

7.3 The Boltzmann transport equation

The Sommerfeld model still ignores the band structure. The Boltzmann transport equation (BTE) merges everything developed so far: electrons live on a band dispersion \(E_n(\mathbf{k})\), move semiclassically according to the acceleration theorem, obey Fermi–Dirac statistics, and scatter.

The description: the phase-space distribution. The semiclassical state of the electron gas here will be described by \(g(\mathbf{r}, \mathbf{k}, t)\), the occupation probability of the state \(\mathbf{k}\) at position \(\mathbf{r}\) and time \(t\) — a distribution over the semiclassical \((\mathbf{r}, \mathbf{k})\) phase space. Its time evolution is ruled by the following equation.

DefinitionThe Boltzmann transport equation

Requiring that \(g\) is conserved along the semiclassical trajectories, except for collisions, means that its convective (total) derivative equals the collision term:

\[\frac{Dg}{Dt} \equiv \frac{\partial g}{\partial t} + \mathbf{v}\cdot\nabla_\mathbf{r}g + \frac{d\mathbf{k}}{dt}\cdot\nabla_\mathbf{k}g = \left(\frac{\partial g}{\partial t}\right)_\text{coll}, \tag{7.9}\]

where the collisional term on the right hand side is in general a complex integral, depending non-linearly on the same phase space distribution, on complicate scattering matrices, etc.

The target: quantify the transport response to external perturbations. In the following, we want to use the BTE to see how the system will respond to external forcing. The response will be first of all a modified \(g(\mathbf{r},\mathbf{k},t)\), but this will be turned into something observable like the charge or heat current densities \(\mathbf{J}\) and \(\mathbf{J}_Q\). Many different forcing terms will be considered, in particular:

• External forces \(\mathbf{F}\), and \(q{\bf E}\) in particular;
• Local temperature gradients, i.e. a generic \(T(\mathbf{r})\) and variation;
• Local density gradients, i.e. a generic \(\mu(\mathbf{r})\) and variation.

The forcing will act on Equation 7.9 via \(\hbar\dot{\mathbf{k}} = \mathbf{F}\) and via distribution variations based on \(T(\mathbf{r})\) and \(\mu(\mathbf{r})\). To proceed towards a useful solution of the problem, a few key approximations have to be made.

DefinitionRelaxation time approximation

The collision term is in general a complicated integral operator. The relaxation time approximation (RTA) replaces it with a simple linear decay towards the local equilibrium,

\[\left(\frac{\partial g}{\partial t}\right)_\text{coll} \approx -\frac{g - g_0}{\tau(\mathbf{k})}, \tag{7.10}\]

with a single (possibly \(\mathbf{k}\)-dependent) relaxation time \(\tau(\mathbf{k})\). The local equilibrium \(g_0\) is the Fermi–Dirac distribution of Equation 7.5, which depends on \(\mathbf{k}\) and \(\mathbf{r}\) only implicitly through \(E(\mathbf{k})\), \(\mu(\mathbf{r})\) and \(T(\mathbf{r})\)

\[g_0(\mathbf{r}, \mathbf{k}) = f(E) = \frac{1}{e^{(E(\mathbf{k})-\mu(\mathbf{r}))/k_BT(\mathbf{r})}+1}. \tag{7.11}\]

DefinitionLinear response approximation

For weak perturbations we write an expansion \(g = g_0 + g_1 + \dots\). Here, \(g_1\) only contains terms linear in the forcing terms, and next terms contain higher-order response, and so on. This allows breaking down the BTE problem into a hierarchy of equations, and in particular

\[\frac{\partial g_1}{\partial t} + \mathbf{v}(\mathbf{k})\cdot\nabla_\mathbf{r}g_0 + \frac{\mathbf{F}(\mathbf{r})}{\hbar}\cdot\nabla_\mathbf{k}g_0 = -\frac{g_1}{\tau(\mathbf{k})}.\]

Note each term is linear in the forcing: \(g_1\) is linear by definition, the \(\nabla_\mathbf{r}g_0\) is proportional to \(\nabla T\) and \(\nabla \mu\), and the last term is linear in \(\mathbf{F}\). Recalling all dependences of \(g_0(\mathbf{r},\mathbf{k}) = f(E)\), chain derivation ruke yields

\[\nabla_\mathbf{k}\,g_0 = -\hbar\,\mathbf{v}(\mathbf{k})\,\left(-\frac{\partial f}{\partial E}\right), \qquad \nabla_\mathbf{r}\,g_0 = \left[\nabla\mu + \frac{E(\mathbf{k})-\mu(\mathbf{r})}{T(\mathbf{r})}\,\nabla T\right]\left(-\frac{\partial f}{\partial E}\right).\]

7.3.1 Solution in the presence of generic constant forcing

To avoid re-doing the same calculation multiple times, here we assume a generic forcing including \(\mathbf{F}\) as well as \(-\nabla T\) and \(-\nabla \mu\). After setting \(\partial_t g_1=0\), simple algebra leads to

\[g_1(\mathbf{r},\mathbf{k}) = \tau(\mathbf{k})\mathbf{v}(\mathbf{k})\cdot\left[\mathbf{F}(\mathbf{r}) - \nabla\mu - \frac{E(\mathbf{k})-\mu(\mathbf{r})}{T(\mathbf{r})}\,\nabla T\right]\left(-\frac{\partial f}{\partial E}\right), \tag{7.12}\]

which is used in calcualtions below. From here on, explicit \(\mathbf{k}\)- and \(\mathbf{r}\)-dependencies will be dropped (!).

CautionConductivity

Let us drop any space dependence and only assume a uniform \(\mathbf{F} = q\mathbf{E}\), with \(q=-e\). We obtain

\[g_1(\mathbf{k}) = q\tau\left(\mathbf{v}\cdot \mathbf{E}\right)\left(-\frac{\partial f}{\partial E}\right).\]

To calculate the net current we only consider \(g_1\) (no net current for \(g_0\))

\[ \mathbf{J} = \frac{2}{(2\pi)^3}\int g_1q\mathbf{v}d^3k =\frac{q^2}{4\pi^3}\int \tau\mathbf{v}\,\left(\mathbf{v}\cdot\mathbf{E}\right)\left(-\frac{\partial f}{\partial E}\right)d^3k, \]

so we obtain

\[(\sigma_0)_{ij} = \frac{q^2}{4\pi^3}\int \tau v_iv_j\left(-\frac{\partial f}{\partial E}\right)d^3k. \tag{7.13}\]

In the limit \(T\to0\), the \((-\partial f/\partial E)\) factor acts as a delta function pinned at the Fermi energy, and the integral only depends on what happens at the Fermi surface. Note that in general \(\sigma\) will be a tensor, but in the presence of sufficiently symmetric bands this will just be a multiple of the identity matrix and practically behave like a scalar. In addition, a real metal will typically contain grains with many orientations, thus averaging out the behavior.

Exercise 1. Verify that if \(E(\mathbf{k})=\hbar^2\mathbf{k}^2/2m\) and \(\tau\) is a constant, we recover the Drude \(\sigma_0=ne^2\tau/m\).

Exercise 2. Assume \(\mathbf{F}(t) = q\mathbf{E}e^{-i\omega t}\) and \(\tau\) as a simple constant. Linearity allows us to write \(g_1(\mathbf{k},t)=g_1(\mathbf{k})e^{-i\omega t}\). Show that this leads to \(\sigma(\omega) = \sigma_0/(1-i\omega\tau)\), reproducing the full functional form of the Drude model.

Show solution

Solution 1. Replacing \(-\partial f/\partial E\) with a Fermi-surface delta, we can make the conductivity a pure Fermi-surface integral

\[(\sigma_0){ij} = \frac{q^2}{4\pi^3}\oint_{E=E_F}\tau\,\frac{v_i v_j}{\hbar|\mathbf{v}|}\,dS, \tag{7.14}\]

which is further simplified in the case of isotropic metal: we can expect the spherical integral of \(v_iv_j\) to be diagonal and coincide to the integral \(v_F^2\,\delta_{ij}/3\) so \(\sigma_0\) becomes a scalar equal to

\[\sigma_0 = \frac{q^2\tau}{4\pi^3}\cdot\frac{v_FS_F}{3\hbar} = \frac{q^2\tau}{4\pi^3}\cdot\frac{4\pi k_F^3}{3m} = \frac{ne^2\tau}{m} \tag{7.15}\]

where \(S_F=4\pi k_F^2\) is the Fermi surface, and replaced \(v_F/\hbar = k_F/m\) and \(n\) with \(2/(2\pi)^3\) times the reciprocal space volume. \(\blacksquare\)

Solution 2. Going back to Equation 7.12 and using \(g_1(\mathbf{k})e^{-i\omega t}\) we have a further term \(\partial_tg_1=-i\omega g_1\), thus we have to replace

\[g_1(\mathbf{k}) = \tau(\mathbf{k})\mathbf{v}(\mathbf{k})\cdot\mathbf{F}\left(-\frac{\partial f}{\partial E}\right) \to \frac{\tau(\mathbf{k})\mathbf{v}(\mathbf{k})\cdot\mathbf{F}}{1-i\omega\tau(\mathbf{r})}\left(-\frac{\partial f}{\partial E}\right)\]

In the limit in which \(\tau\) can be regarded as a simple constant, we just have an extra factor \(1/(1-i\omega\tau)\). \(\blacksquare\)

The integral in Equation 7.13 is recurrent in the transport equations and merits the specific definition

\[K_n = \frac{1}{4\pi^3} \int \tau v_iv_j(E-\mu)^n\left(-\frac{\partial f}{\partial E}\right)d^3k, \tag{7.16}\]

which will be called transport coefficient. Now we can just write \(\sigma_0=e^2K_0=q^2K_0\). Higher order coefficients \(K_n\) become relevant when considering thermal effects and heat transport.

CautionSeebeck effect

Let us now consider the effect of a forcing \(\nabla T\), the linear response to \(\nabla T\) in Equation 7.12 is completely equivalent to one to the force \(\mathbf{F}=q\mathbf{E}\), except that the factor \(q\) is replaced by \(-(E-\mu)/T\); this shifts \(qK_0\to -K_1/T\). We can thus immediately write

\[\mathbf{J} = -\frac{qK_1}{T}\nabla T = q^2K_0\left(-\frac{K_1}{qK_0<t}\nabla T\right)\]

which we is normally absorbed into the combined equation \(\mathbf{J} = \sigma(\mathbf{E}-S\nabla T),\) where we introduced the Seebeck coefficient \(S=K_1/qK_0T\). Remember both \(K_0\) and \(K_1\) are in general tensors.

Signs. Here \(q=-e\) is negative, \(K_0\) is expected to be positive, and \(K_1\) can be positive or negative depending on whether the factor \(v_iv_j\) in the integral grows or not with energy. While the naive expectation for a metal is \(K_1>0\) and \(S<0\), it all depends on the band structure. At the top of a band, \(K_1\) can easily turn negative (or, equivalently, we can switch to the hole picture where \(q=+e\)).

CautionDiffusion phenomena

Let us finally consider the effect of a density gradient. At uniform temperature this is encoded in \(\mu(\mathbf{r})\), and in Equation 7.12 the drive \(-\nabla\mu\) plays the same role of the force \(\mathbf{F} = q\mathbf{E}\): repeating the conductivity calculation we thus have

\[\mathbf{J} = qK_0\,\left(-\nabla\mu\right).\]

Since we normally want to write these equations in terms of \(\nabla n\), we can calculate

\[\nabla\mu = \left(\frac{\partial \mu}{\partial n}\right)_{\!T} \nabla n,\]

and thus write

\[\mathbf{J} = -qK_0\left(\frac{\partial \mu}{\partial n}\right)_{\!T} \nabla n = -qD\,\nabla n,\]

where we introduced the diffusion coefficient \(D\). It is instructive to calculate \(D\): here we consider the two limit cases of the Sommerfeld model for \(T\to0\) and of a classic gas with Boltzmann distribution:

• Boltzmann limit. Here \(n\propto \exp\left(\mu/k_BT\right)\) and \(\partial n/\partial \mu = n/k_BT\) so (Einstein relation)

\[D = K_0\frac{\partial \mu}{\partial n} = \frac{n\mu_e}{e}\frac{k_BT}{n} = \mu_e\frac{k_BT}{e}.\]

• Sommerfeld limit. Here \(\mu \approx E_F\propto k_F^2 \propto n^{2/3}\) so we have \(\partial \mu/\partial n = 2E_F/3n\) and thus

\[D = K_0\frac{\partial \mu}{\partial n} = \frac{n\mu_e}{e}\frac{2k_BT_F}{3n} = \frac{2}{3}\mu_e\frac{k_BT_F}{e}.\]

Note the two limits differ by a factor \(T_F/T\) plus a minor variation in the numerical factor in front. Again statistics has a deep impact and strongly enhances \(D\) for a given mobility.

CautionMetal comparison: why do some metals conduct better than others?

We have already noted that in a metal with isotropic bands the conductivity can be written as

\[\sigma_0\frac{q^2\tau v_F S_F}{12\pi^3\hbar}\]

where \(S_F\) is the Fermi surface of the metal. Written in this form, the formula carries a clear message: the conductivity of a metal is controlled by three Fermi-surface properties, more than by the total electron density: the area \(S_F\) of the Fermi surface, the velocity \(v_F\) of the states on it, and the relaxation time \(\tau\) of those same states. This immediately explains the hierarchy observed among real metals:

• Noble metals (Cu, Ag, Au) are the best conductors. Their single \(s\) valence electron produces a large, nearly-free-electron spherical Fermi surface with high Fermi velocities (\(v_F \sim 1.5\times10^6\) m/s); the filled \(d\) shells lie well below \(E_F\), so the density of states at the Fermi level is low and there are few final states available for scattering – long \(\tau\). Large \(S_F\), large \(v_F\), long \(\tau\): all three factors are simultaneously favorable.
• Transition metals conduct worse despite having more valence electrons. The Fermi level crosses the narrow \(d\) bands: these are flat, so their velocities are small, and their enormous density of states opens a very efficient scattering channel for the fast \(s\) electrons, which shortens \(\tau\).
• Semimetals tend to conduct poorly simply because their Fermi surface is tiny (\(S_F \to 0\)), no matter how clean they are.

In short: a good metal is one with a large Fermi surface populated by fast electrons that have nowhere to scatter.

7.4 Full linear transport equations

When an electric field, a carrier-density gradient and a temperature gradient are all present, the charge current \(\mathbf{J}\) and the heat current \(\mathbf{J}_Q\) are described by two coupled linear response equations:

\[\begin{aligned} \mathbf{J} \;&=\; \sigma(\mathbf{E} -S\nabla T)- qD\,\nabla n,\\ \mathbf{J}_Q \;&=\; \Pi\,\mathbf{J} \;-\; \kappa_e\,\nabla T, \end{aligned} \tag{7.17}\]

where we introduce \(\Pi = TS\) the Peltier coefficient and \(\kappa_e\) the thermal conductivity of electrons (measured at zero charge current). A key message is that moving charge always corresponds to moving heat, according to the magnitude of \(\Pi\). Note that in a real crystal we expect a total thermal conductivity \(\kappa = k_e+k_{ph}\), including also heat transport mediated by phonons.

CautionFull derivation

Here we still miss the calculation of \(\mathbf{J}_Q\), which can be relatively easily extrapolated from previous results. First of all we have to clarify what is a heat current. Heat current is different from energy current and the difference can be clearly understood from thermodynamics

\[dU = dQ+\mu dN,\]

indicating that energy change can be caused by heat exchange (first term) or particle exchange (second term). Accordingly, energy current density \(\mathbf{J}_E\), heat current density \(\mathbf{J}_Q\) and particle current density \(\mathbf{J}_N\) are related by

\[\mathbf{J}_E = \mathbf{J}_Q+\mu\mathbf{J}_N,\]

where thus when calculating \(\mathbf{J}_Q = \mathbf{J}_E-\mu\mathbf{J}_N\) the transported quantity is not anymore \(q\) but \((E-\mu)\), which we have to replace in all the current density integrals. This leads to a set of mappings, in particular: \(qK_0\to K_1\) and \(qK_1\to K_2\). Thus we have

\[ \begin{aligned} \mathbf{J}_Q &= qK_1\mathbf{E}-\frac{K_2}{T}\nabla T - D\frac{K_1}{K_0}\nabla n\\ &= \frac{K_1}{qK_0}\left(e^2K_0\mathbf{E}-q\frac{K_1}{T}\nabla T - qD\nabla n\right) - \left(\frac{K_2}{T}-\frac{K_1^2}{TK_0}\right)\nabla T\\ &= \Pi\mathbf{J} - \kappa\nabla T \end{aligned} \]

where \(\Pi=ST=K_1/qK_0\) and \(\kappa = K_2/T-K_1^2/TK_0\).

CautionThe spectral density of the transport coefficients

Here we want to take advantage of Sommerfeld expansion in the study of the transport equation. To this end, the coefficients \(K_n\) have to be converted into energy integrals. This can be done by introducing the spectral density of the transport coefficients

\[\Sigma(E) = \frac{1}{4\pi^3}\int \tau(\mathbf{k})v_i(\mathbf{k})v_j(\mathbf{k})\delta(E-E(\mathbf{k}))d^3k\]

so that we can write

\[K_n = \int (E-\mu)^n\,\Sigma(E)\left(-\frac{\partial f}{\partial E}\right)dE.\]

which can be easily expanded using the integral form of Theorem 7.1

\[\int G(E)\left(-\frac{\partial f}{\partial E}\right)dE = G(\mu) + \frac{\pi^2}{6}(k_BT)^2 G''(\mu) + \mathcal{O}(T^4).\]

Charge conductivity

Here we need to compute \(K_0\) using \(G(E)=\Sigma(E)\); the integral can be expanded (\(\mu\approx E_F\)) as

\[\sigma_0=e^2K_0 = e^2\Sigma(\mu) +\mathcal{O}(T^2).\]

Seebeck coefficient

Here we need also \(K_1\) and \(G(E)=(E-\mu)\Sigma(E)\) which vanishes at \(\mu\) so the leading term comes from the second derivative

\[K_1 = \frac{\pi^2}{6}(k_BT)^2\frac{d^2}{dE^2}\left[(E-\mu)\sigma(E)\right] = \frac{\pi^2}{3}(k_BT)^2\,\Sigma'(\mu)+\mathcal{O}(T^4)\]

and we can calculate the leading term

\[S = \frac{K_1}{qTK_0} = -\frac{\pi^2}{3}\,\frac{k_B}{e}\,\frac{\Sigma'(\mu)k_BT}{\Sigma(\mu)}.\]

Thermal conductivity

Here we need also \(K_2\) and \(G(E)=(E-\mu)^2\,\Sigma(E)\). Now both value and first derivative vanish at \(\mu\), so we get

\[K_2 = \frac{\pi^2}{3}(k_BT)^2\,\Sigma(\mu)+\mathcal{O}(T^4)\]

and finally the leading term

\[\kappa = \frac{K_2}{T}-\frac{K_1^2}{TK_0} \approx \frac{K_2}{T} = \frac{\pi^2}{3}\,k_B^2\,T\,\Sigma(\mu),\]

where the \(K_1^2/K_0\) correction is smaller by a factor \(\sim(k_BT/E_F)^2\) and was neglected.

Wiedemann–Franz law. Taking the ratio of the two leading terms, \(\Sigma(\mu)\), and with it every detail of the band structure and of the scattering, cancels out and we obtain the universal Lorenz number

\[\frac{\kappa}{\sigma_0\,T} = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 \equiv L_0 \approx 2.44\times10^{-8}\ \mathrm{V^2\,K^{-2}}. \tag{7.18}\]

In conclusion…

ImportantTake home messages…

At the end of this chapter you should know…

• Drude model. Starting from the basics of Drude theory learned in electrodynamics courses (you should recall them)… what is dissatisfactory in the Drude model?
• Sommerfeld model. The Fermi sea and orders of magnitude (\(k_F\), \(E_F\), \(T_F\)); the Sommerfeld expansion and its basic applications. Does this solve the problem above?
• Boltzmann transport equation. How do we add band dispersions to the discussion? Definition of the BTE and approximations: RTA, linear response. Derivation of conductivity.
• Linear transport equations What rules linear charge and heat response in a crystal? Key phenomenologies: conduction, Seebeck effect, diffusion (Fick’s law), Peltier effect, thermal conductivity (Fourier’s law).